Schedule

• September 11

  Yier Lin
  University of Chicago

  Multi-point Lyapunov Exponents of the Stochastic Heat Equation

  Abstract: We study the Stochastic Heat Equation with multiplicative space-time white noise. Extensive research has already been conducted on the one-point Lyapunov exponents of this equation. In this talk, I will present how we can compute the multi-point Lyapunov exponents by leveraging a combination of integrability and probability. As a byproduct, we also solve a quadratic optimization problem.

• September 18

  Lingfu Zhang
  UC Berkeley

  Geodesics in Last-Passage Percolation under Large Deviations

  Abstract: In KPZ universality, an important family of models arises from 2D last-passage percolation (LPP): in a 2D i.i.d. random field, one considers the geodesic connecting two vertices, which is defined as the up-right path maximizing its weight, i.e., the sum/integral of the random field along it. A characteristic KPZ behavior is the 2/3 geodesic fluctuation exponent, which has been proven for some LPPs with exactly solvable structures. A topic of much recent interest is such models under upper- and lower-tail large deviations, i.e., when the geodesic weight is atypically large or small. In prior works, it was established that the geodesic exponent changes to 1/2 (more localized) and 1 (delocalized) respectively. In this talk, I will describe a further refined picture: the geodesic converges to a Brownian bridge under the upper tail, and a uniformly chosen function from a one-parameter family under the lower tail. I will also discuss the proofs, using a combination of algebraic, geometric, and probabilistic arguments.

  This is based on two forthcoming works, one joint with Shirshendu Ganguly and Milind Hegde, and the other with Shirshendu Ganguly.

• September 25

  Amol Aggarwal
  Columbia University

  A Characterization for the Airy Line Ensemble

  Abstract: The Airy line ensemble is a universal scaling limit that is believed (and in some cases proven) to govern the fluctuations of many probabilistic systems, such as random surfaces, interacting particle systems, and stochastic interfaces. It is an example of a "Brownian line ensemble," which informally means that it is an infinite, ordered sequence of random continuous curves that look like non-intersecting Brownian motions. In this talk we survey recent results characterizing the Airy line ensemble as the unique Brownian line ensemble whose top curve decays parabolically, and we explain why this result is useful for proving convergence theorems for various discrete stochastic models. This is based on joint work with Jiaoyang Huang.

• October 2

  Morris Ang
  Columbia University

  Cutting LQG by SLE with a mismatched central charge

  Abstract: There are many deep and useful theorems relating Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG) when their parameters are matched, meaning $\kappa \in \{ \gamma^2, 16/\gamma^2\}$. Roughly speaking, the SLE curve cuts the LQG surface into two or more independent LQG surfaces. We extend these theorems to the setting of mismatched parameters: an LQG disk is cut by an SLE curve into two or more LQG surfaces which are conditionally independent given the values along the SLE curve of a certain collection of auxiliary imaginary geometry fields. These fields are sampled independently of the LQG and SLE, and have the property that the central charges of the LQG, SLE and auxiliary fields sum to 26. This central charge condition is natural from the perspective of bosonic string theory. Similar statements hold when the SLE curve is replaced by, e.g., an LQG metric ball or a Brownian motion path. These statements are continuum analogs of certain Markov properties of random planar maps decorated by two or more statistical physics models. This work is joint with Ewain Gwynne.

• October 9

  Indigenous Peoples' Day

• October 16

  Ramon von Handel
  Princeton

• Octobe 23

  Hao Shen
  UW Madison

• October 30

  
  

• November 6

  Jacopo Borga
  Stanford

• November 13

  Zhongyang Li
  University of Connecticut

• November 20

  Bjoern Bringmann
  IAS/Princeton

• November 27

  
  

• December 4

  
  

• December 11

  Michael Magee
  Durham University & IAS